Let's understand the properties of a Relation on a set A. A Relation R on a set A is a subset of \(A \times A\). The question asks about a specific property of a Relation R where R is equal to its inverse relation, denoted by R^-1.

Understanding the Inverse Relation (R^-1)

The inverse Relation R^-1 is defined based on the original Relation R. If an ordered pair \((a, b)\) is in R, then the ordered pair \((b, a)\) is in R^-1. Mathematically:

\[(b, a) \in R^{-1} \iff (a, b) \in R\]

Analyzing the Condition R = R^-1

The given condition is R = R^-1. This means that the set of ordered pairs in R is exactly the same as the set of ordered pairs in R^-1. Let's see what this implies.

If R = R^-1, then for any ordered pair \((a, b)\):

• If \((a, b) \in R\), then since R = R^-1, we must have \((a, b) \in R^{-1}\).
• According to the definition of R^-1, if \((a, b) \in R^{-1}\), then \((b, a) \in R\).

So, the condition R = R^-1 implies that if \((a, b) \in R\), then \((b, a) \in R\). This is a specific property of relations.

Comparing R=R^-1 with Properties of Relations

Let's look at the standard properties of relations: Reflexive, Symmetric, and Transitive.

Reflexive Relation

A Relation R on a set A is Reflexive if for every element \(a \in A\), the ordered pair \((a, a)\) is in R. That is, \(\forall a \in A, (a, a) \in R\). The condition R = R^-1 does not require that \((a, a)\) must be in R for all \(a \in A\). For example, if A = {1, 2} and R = {(1, 2), (2, 1)}, then R^-1 = {(2, 1), (1, 2)}, so R = R^-1, but R is not Reflexive because \((1, 1) \notin R\) and \((2, 2) \notin R\).

Symmetric Relation

A Relation R on a set A is Symmetric if whenever the ordered pair \((a, b)\) is in R, the ordered pair \((b, a)\) is also in R. That is, if \((a, b) \in R\), then \((b, a) \in R\). We derived this exact condition from R = R^-1.

If R = R^-1, then:

Let \((a, b) \in R\).
Since R = R^-1, this means \((a, b) \in R^{-1}\).
By the definition of the inverse relation, if \((a, b) \in R^{-1}\), then \((b, a) \in R\).

Thus, if \((a, b) \in R\), then \((b, a) \in R\). This matches the definition of a Symmetric relation.

Transitive Relation

A Relation R on a set A is Transitive if whenever \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\). The condition R = R^-1 does not guarantee this property. For example, if A = {1, 2, 3} and R = {(1, 2), (2, 1), (2, 3), (3, 2)}, then R^-1 = {(2, 1), (1, 2), (3, 2), (2, 3)}, so R = R^-1. However, R is not Transitive because \((1, 2) \in R\) and \((2, 3) \in R\), but \((1, 3) \notin R\).

Conclusion

The condition R = R^-1 directly corresponds to the definition of a Symmetric Relation. If a Relation R is equal to its inverse, it means that for every pair \((a, b)\) in the relation, the reversed pair \((b, a)\) is also in the relation, which is exactly what being Symmetric means.